The Tortoise challenged Achilles to a race, claiming that he would win as long as Achilles gave him a small head start. Achilles laughed at this, for of course he was a mighty warrior and swift of foot, whereas the Tortoise was heavy and slow.
“How big a head start do you need?” he asked the Tortoise with a smile.
“Ten meters,” the latter replied.
Achilles laughed louder than ever. “You will surely lose, my friend, in that case,” he told the Tortoise, “but let us race, if you wish it.”
“On the contrary,” said the Tortoise, “I will win, and I can prove it to you by a simple argument.”
“Go on then,” Achilles replied, with less confidence than he felt before. He knew he was the superior athlete, but he also knew the Tortoise had the sharper wits, and he had lost many a bewildering argument with him before this.
“Suppose,” began the Tortoise, “that you give me a 10-meter head start. Would you say that you could cover that 10 meters between us very quickly?”
“Very quickly,” Achilles affirmed.
“And in that time, how far should I have gone, do you think?”
“Perhaps a meter—no more,” said Achilles after a moment’s thought.
“Very well,” replied the Tortoise, “so now there is a meter between us. And you would catch up that distance very quickly?”
“Very quickly indeed!”
“And yet, in that time I shall have gone a little way farther, so that now you must catch that distance up, yes?”
“Ye-es,” said Achilles slowly.
“And while you are doing so, I shall have gone a little way farther, so that you must then catch up the new distance,” the Tortoise continued smoothly.
Achilles said nothing.
“And so you see, in each moment you must be catching up the distance between us, and yet I—at the same time—will be adding a new distance, however small, for you to catch up again.”
“Indeed, it must be so,” said Achilles wearily.
“And so you can never catch up,” the Tortoise concluded sympathetically.
“You are right, as always,” said Achilles sadly—and conceded the race.
This is actually sequences and series that leads to an important mathematical concept that I studied in the college classroom and was tested on. Did you take those classes? Did you read the book? Here the answer is in this book. . . — Jeremiah
When I was a kid, I was taught, like Jeremiah here, that limits and convergent series and calculus "solve" Zeno's paradox. — Srap Tasmaner
What's curious is that even in a high school science class there's likely a kid who'll argue that you can't subdivide matter infinitely -- everyone's heard something about particles, even if it's hard to understand. But the idea that space is granular just seems crazy! — Srap Tasmaner
What's curious is that even in a high school science class there's likely a kid who'll argue that you can't subdivide matter infinitely -- everyone's heard something about particles, even if it's hard to understand. But the idea that space is granular just seems crazy! — Srap Tasmaner
While it can be divided an infinite number of times, it still sums up to a whole, hence we can cross the room — Jeremiah
Ask yourself: How is that you can have an infinite number of distances to cover, but yet still be able to cross the room? — Jeremiah
Sure it does, Zeno's paradox is about getting from point a (the start of the race) to point b (the turtle) over infinite small intermediate distances. This is exactly what calculus is. — Jeremiah
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